SCIENTIFIC PROGRAMS AND ACTIVITIES

December 9, 2016

Seminar Series on Control Theory

Date: June 4, 1992

Topic: "Feedback Equivalence to Linear Prime Systems"

Speaker: Ricardo Marino, University of California, Santa
Barbara

State space and output space change of coordinates and static state
feedback are used to transform a multivariable nonlinear system with
m-inputs and m-outputs into a linear prime system, i.e. m chain of integrators
from the new outputs to the new inputs. Necessarky and sukfficient differential
geometric conditions for the solvability of the problem are given. They
generalize well-known results both on feedback linearization as well
as on input output decoupling of nonlinear systems. From a computational
point of view the output space change of coordinates is the crucial
step, which is performed by constructing rectifying coordinates for
a nested sequence of distributions on the output manifold.

Date: June 4, 1992

Speaker: Ricardo Marino, University of California, Santa
Barbara

Differential geometric conditions are given to identify a class of
single-input, single-output nonlinear systems which are globally transformable
by dynamic output feedback control, "filtered' state space change
of coordinates and output space change of coordinates into a linear
observable minimum phase system. A global output tracking control can
be designed for those systems. It is then shown that some conditions
may be relaxed and a global output tracking course can still be designed
without relying on the dynamic output feedback linearization property.

Date: May 15, 1992

Topic: "Elastic Curves and Related Variational Problems"
- The Second of Two Talks

Speaker: Velimir Jurdjevic, Toronto and Fields Institute

Many classical variational problems with constraints, when viewed as
optimal contol problems can be effectively analyzed through the Maximum
Principle and its associated Hamiltonian formalism. In this lecture
I will discuss the equilibrium configurations of a thin elastic, inextensible,
rod in R^3, by treating it as a left-invariant variational problem on
the group of motions of R^3. I will divide the equations of motion,
show the connection with the equations of the rigid body (kinetic analogue
of Kirchhoff), and discuss the integrability of the resulting systems.

Date: May 13, 1992

Topic: "Elastic Curves and Related Variational Problems"
- The First of Two Talks

Speaker: Velimir Jurdjevic, Toronto and Fields Institute

Many classical variational problems with constraints, when viewed as
optimal contol problems can be effectively analyzed through the Maximum
Principle and its associated Hamiltonian formalism. In this lecture
I will discuss the equilibrium configurations of a thin elastic, inextensible,
rod in R^3, by treating it as a left-invariant variational problem on
the group of motions of R^3. I will divide the equations of motion,
show the connection with the equations of the rigid body (kinetic analogue
of Kirchhoff), and discuss the integrability of the resulting systems.

Date: May 8, 1992

Topic: "Time Control of Tidal Generation"

Speaker: G.F.D. Duff, University of Toronto

Because of its almost periodic intermittent and time precessing nature,
the availability of electric power from turbines in a tidal barrier
requires time dependent controls. The nature of this problem will be
described, together with some of the elaborations of design and operation
that are possible. Systems with single, double or multiple basins are
considered, with reference to the problem of construction cost.

Mathematical formulation by way of optimal control theory or by linear
programming is described, and interaction with the hydrodynamical wave
properties of the ocean system included. Numerical calculations, with
references to stability and the frequent occurrence of focal points
which act as limiting factors, will be described, and specific models
based on the sites in Nova Scotia and New Brunswick discussed.

Speaker: Ron Perline, Drexel University, Philadelphia

Date:April 22, 1992

Speaker: Nicholas Varpoulos, Pierre et Marie Paris IV

Date: April 10, 1992

Topic: "Global Properties of Compact Curves and
Surfaces in Minkowski 3-space" - The Second of Two Talks

Speaker: Marek Kossowski, University of South Carolina

The classical global theorems for curves and surfaces in Euclidean
3-space have analogies in Minkowski 3-space if the curves and surfaces
satisfy certain generic conditions. The talk will give an overview.

Speaker: Peter Crouch, Arizona State

Date: March 23, 1992

Speaker: Mohammed Dahleh, University of California, Santa
Barbara

Date: March 20, 1992

Topic: "Smoke Rings and Springy Wires"

Speaker: Joel Langer, Case Western Reserve University

The vortex filament equation (VF) is a model for the evolution of a
thin vortex filament in three dimensional hydrodyanmics. This model
is known to be a completely integrable infinite dimensional Hamiltonian
system, and it is also known that equilibrium configurations of the
Kirchhoff elastic rod model yield soliton solutions of (VF). The latter
model is also an integrable Hamiltonian system (finite dimensional).
In the talk, relationships between the two integrable systems will be
discussed.

Date: March 13, 1992

Speaker: Parick McDonald, Ohio State University

Date: March 6, 1992.

Topic: "Local Invariants in Sub-Riemannian Geometry"

Speaker: George Wilkens, University of Hawaii

In sub-Riemannian geometry, one is given a smooth manifold, M, and
a smooth, constant rank subbundle, D, of the tangent bundle of M. We
assume that D satisfies the bracket generating condition, that
is that at each point x in M, the tangent space to M at x (i.e. TxM)
is spanned by the vectors in the Lie algebra generated by the vector
fields of M which lie in D. Additionally, one is given a smooth function
a: D -> R such that the restriction of a to each fiber is a positive
definite quadratic form. With this structure, we say that a smooth curve
in M is an integral curve of D if x'(t) lies in D for all t.We define
an energy functional on integral curves by E(x) = 1/2 Integrate [a(x'(t))
dt]. Sub-Riemannian geometry is the geometry associated to the distribution
D and the energy E.

I will take M to be a three dimensional manifold and D to be a rank
two subbundle of TM. I will show that one can use geometrically nautral
conditions to find all the local invariants of this structure. There
are four fundamental invariants. These invariants all vanish if and
only if the geometry is the one described by the Heisenberg fly wheel.
Examples with three dimensional Lie groups of symmetries are easy to
pick out, and the symmetry groups include semi-simple, solvable and
nillpotent Lie groups. There is a related two dimensional manifold N
with an invariantly defined volume form. This volume form describes
part of the motion of integral curves. Special cases involving metrics
on the surface are easily identified.

Date: February 28, 1992

Topic: "Robust Control on Metric Spaces of Systems"

Speaker: Li Qiu, Postdoctoral Fellow, Fields Institute

Robust control deals with the control of uncertain systems, or equivalently
sets of systems. An uncertain system is usually described as a neighborhood
of a fixed nominal system. Such a neighborhood can therefore be defined
through a metric in the space of all systems. In this talk, several
such metrics in the space of linear time-invariant systems will be introduced
and an overview of a robust control theory based on these metrics will
be given.

Date: February 21, 1992

Topic: "New Integrable Problem of Classical Mechanics"

Speaker: Oleg Bogoyavlenskij, Steklov Mathematical Institute,
Moscow

The talk will be devoted to the proof that the dynamics of an arbitrary
rigid body in a gravitational field with an abitrary quadratic potential
is completely integrable in Liouville's sense. The dynamics of the centre
of mass of the rigid body is integrable in elementary functions; rotation
of the rigid body around its centre of mass is integrable in theta-functions
on Riemannian surface.

Date: February 14, 1992

Topic: "Centro-Affine Plane Curves and Feedback
Control "

Speaker: George Wilkens, University of Hawaii

Date: February 14, 1992

Topic: "Geometric Phases and the Control of Super-Articulated
Mechanical Systems"

Super-articulated mechanical systems have more degrees of freedom than
actuators. For these systems, the relationship between actuator inputs
and configuration space trajectories is nonholonomic in the sense that
when the input variables return to their intial values it will not generally
be the case that configuration variables return to their initial values.
In this talk, the long-term effects produced by periodic forcing of
super-articulated mechanical systems are studied. For Lagrangian control
systems with symmetry, it is shown that the stability of equilibrium
motions may be assessed in terms of a quantity which we call the average
potential. Although the definition of this may be motivated by classical
averaging theory, we shall argue that it is essentially a geometric
invariant.